1 Answer
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Yes, you could connect a switch to the router then connect as many computer as you have switch ports remaining.

For example

A router with 4 LAN ports, could have one 24 port switch and 3 computers connected to it.
The switch would then be able to connect 23 computers to it. So, in this scenario you could have 26 computer connected.

Likely the router is setup as the DHCP server. If you have the login information you can access it by typing its IP address in a web browsers address bar. Usually the IP address is defaulted to 192.168.1.1 or 192.168.1.0. If you already have a computer connecting to the router and getting its IP address via DHCP you can do the following to find the routers IP address:

Click on Start

search for cmd

click on cmd to open command prompt

type ipconfig and hit enter

The default gateway will be your router

Once you log into the router and go to where you can view the DHCP settings you will see what the range is. This is how many computers you can have connected. You can change the range. With IPv4 You can get a max of 254 address on one scope. Since the router si using one of those addresses you are left with 253.

To make sure your computers are getting there IP addresses via DHCP perform the following:

Go to Control Panel

View network status and tasks

Change Adapter settings

Right click on your connection and select Properties

Select Internet Protocol Version 4 (or 6, depends on which one you
are using)

Click Properties

Make sure that you are obtaining the IP addresses automatically

As long as you are using an unmanaged switch you will not have to do anything to configure the switch. I would not use multiple switches connected to each other. If you are going to use multiple switches then connect them all to the router directly.